How currency affects a return you already have

Buying a foreign asset is two investments. Most people make one of them deliberately.

The decomposition

Write $R_{\text{local}}$ for the asset's return in its own currency and $c$ for the change in the foreign currency's value against the rupee. Then your rupee return is

$1 + R_{\text{INR}} = \left(1 + R_{\text{local}}\right)\left(1 + c\right)$

They multiply. Expanding:

$R_{\text{INR}} = R_{\text{local}} + c + R_{\text{local}} \times c$

Two legs and a cross term. The cross term is small for small moves and is not small when either leg is large — a 20% asset gain with a 10% currency gain adds a further 2 points on top of the 30.

This is an identity, not a model. No assumption is involved. Whatever the asset does and whatever the rate does, those are the only two sources of your return.

What the combinations look like

Rupee returns, for a range of asset returns and currency moves:

Asset return	Currency −10%	−5%	0%	+5%	+10%
−20%	−28.0%	−24.0%	−20.0%	−16.0%	−12.0%
−10%	−19.0%	−14.5%	−10.0%	−5.5%	−1.0%
0%	−10.0%	−5.0%	0.0%	+5.0%	+10.0%
+10%	−1.0%	+4.5%	+10.0%	+15.5%	+21.0%
+20%	+8.0%	+14.0%	+20.0%	+26.0%	+32.0%

Read the middle row first. With the asset doing nothing at all, the rupee return ranges from −10% to +10%. The currency alone can produce the whole of a year's result.

And read the corners. The same asset return of +10% delivers −1.0% or +21.0% depending on the currency. That is a 22-point spread on an identical investment decision, determined by something most buyers of an international fund never formed a view on.

Why this is not symmetric for an Indian investor

A reader might conclude the currency is a coin flip that averages out. Over long periods it has not been, and the reason is structural rather than lucky.

India has generally run higher inflation than the economies whose assets Indians buy. Chapter 1's real-exchange-rate argument then implies a tendency for the rupee to depreciate against those currencies over long horizons, roughly tracking the inflation gap.

So a foreign holding has had a long-run tailwind from the currency, and an Indian investor buying US assets has historically been paid twice for being right once.

Three cautions on that before anyone treats it as a free lunch.

The tailwind is the mirror of your own inflation. If the rupee falls 4% a year because Indian prices rise 4% faster, the currency gain is restoring purchasing power you were losing anyway. In real terms it is not a gain. The Measuring your return subject's chapter on real returns is the one that matters here.

It is a tendency, not a rule, and it reverses. The table above includes a −10% currency column because rupee appreciation happens.

And it is already in the price of hedging. Chapter 5 shows that the forward rate prices the rate differential, so a hedged investor gives up precisely this expected drift. You cannot have the depreciation tailwind and the protection at once — that is the trade, and it is priced.

What it does to risk

Returns multiply, and so, approximately, do the risks compound. For small moves:

$\sigma_{\text{INR}}^2 \approx \sigma_{\text{local}}^2 + \sigma_c^2 + 2\rho\,\sigma_{\text{local}}\,\sigma_c$

That is chapter 2 of the Portfolio theory subject, applied to two legs of one holding. And the correlation term decides whether the currency adds risk or removes it.

The case for foreign assets reducing portfolio risk rests entirely on that correlation. Indian equities and the rupee tend to fall together in a crisis — foreign money leaves, selling both. A foreign asset held unhedged gains in rupee terms exactly when Indian assets are falling, which is a genuinely valuable property and the strongest argument in this subject.

The honest qualification is the one the Risk subject makes about all such relationships: it is measured in ordinary conditions, estimated with error, and may not hold in the episode you are relying on it for.

And note the direction of the design implication. If the reason to hold foreign assets is this diversification, then hedging the currency removes the reason — because the hedge removes exactly the leg that moves against Indian assets. Chapter 5 is where that decision gets made.

The same thing from the other side

NSE publishes the Nifty 50 in variants including Nifty50 USD and Nifty50 JPY — the same index, measured in different currencies.

Those variants exist because the question runs both ways. A foreign investor in Indian equities faces exactly this decomposition with the signs reversed: the Nifty's 6.38% five-year rupee return is a different number in dollars, and the difference is the rupee's path over those five years.

Which is a useful discipline to borrow. When you read any cross-border return, ask what currency it is stated in. "The fund returned 12%" is an incomplete sentence in this subject, in the same way that "this asset is risky" was incomplete in Portfolio theory.

Working the problem

Invested at 90 rupees per dollar. The fund rose 10% in dollars. The rate is now 96.6149.

Step 1 — the currency leg. The rate moved from 90 to 96.6149, so each dollar now converts to more rupees:

$c = \frac{96.6149}{90} - 1 = +7.35\%$

Step 2 — the asset leg. $R_{\text{local}} = +10\%$, given.

Step 3 — combine, multiplying.

$1 + R_{\text{INR}} = 1.10 \times 1.0735 = 1.1808$

$R_{\text{INR}} = +18.08\%$

Step 4 — split it.

Source	Contribution
The asset	10.00 points
The currency, including the cross term	8.08 points
Total	18.08 points

Nearly 45% of the return came from the rupee falling.

Step 5 — which one you had a view on. Almost certainly the asset. You chose a fund because you had some opinion about US equities, or about diversifying away from India, or because it had done well.

You took the currency position as a by-product. It was not sized, not compared against alternatives, and not deliberately held — and it supplied nearly half the outcome.

Three things follow, and they are the practical content of this chapter.

It could have gone the other way, and by a similar amount. Had the rupee strengthened from 90 to 85, the same 10% fund return would have delivered

$1.10 \times \frac{85}{90} - 1 = +3.89\%$

Same fund, same year, 3.89% instead of 18.08% — a 14-point swing from the leg nobody chose.

The currency can be the entire story. If the fund had been flat, the 7.35% currency move would have been the whole of your return. The table's middle row is not a hypothetical.

And the right response is to decide rather than to default. Chapter 5 sets out the choice between hedged and unhedged, and the point of this chapter is that not deciding is also a decision — it is an unhedged position, chosen by inattention. For an investor whose purpose is diversification away from Indian assets, unhedged is usually the right answer; the fault is in holding it without knowing.

The point

A foreign holding's rupee return is $(1 + R_{\text{local}})(1 + c) - 1$, so the two legs multiply and carry a cross term. The currency alone can produce a full year's result: with the asset flat, a move from 90 to 96.6149 rupees per dollar is a 7.35% return, and the same 10% fund gain becomes 18.08% or 3.89% depending on the rate. India's higher inflation has given foreign holdings a long-run currency tailwind, which is largely the mirror of domestic inflation and which is already priced into the cost of hedging. The currency leg also adds or removes risk through its correlation with Indian assets — and because the rupee tends to fall when Indian equities fall, the unhedged currency exposure is the main reason foreign assets diversify an Indian portfolio at all.